On the uniqueness of meromorphic functions that share small functions on annuli

1.   Introduction and main result

In this article, we assume that the readers are familiar with the basic results and the standard notations of Nevanlinna's value distribution theory ^[16,18]. Let $f$ and $g$ be two non-constant moromorphic functions, and let $a$ be a complex number or a small function with respect to $f$ and $g$. Then, we say that $f$ and $g$ share $a$ IM (or CM) provided that $f-a$ and $g-a$ have the same zeros ignoring (or counting) multiplicities.

It is well known that R. Nevanlinna ^[11] proved the five-value theorem in 1926: For two non-constant meromorphic functions $f$ and $g$ in the complex plane $\mathbb{C}$, we have $f\equiv g$ providing that $f$ and $g$ share $a_{j}$ IM for $j = 1, 2, 3, 4, 5,$ where $a_{j} (j = 1, 2, 3, 4, 5)$ are five distinct values. In 2000 and 2001, Y. H. Li, J. Y. Qiao ^[9] and H. X. Yi ^[17] extended this very work to the case of sharing five small functions, proving the five small functions theorem: Let $f$ and $g$ be two non-constant meromorphic functions in the complex plane $\mathbb{C}$, and $a_{j} (j = 1, 2, 3, 4, 5)$ be five distinct small functions with respect to $f$ and $g$. If $f$ and $g$ share $a_{j} (j = 1, 2, 3, 4, 5)$ IM in $\mathbb{C}$, then $f\equiv g$. In 2003 and 2004, J. H. Zheng ^[19,20] proved the five value theorem in one angular domain. In 2011, H. F. Liu and Z. Q. Mao ^[10] further gave the five small functions theorem in one angular domain.

Next we will mainly discuss the uniqueness theory of meromorphic functions on annuli. For the basic results and necessary notations as $T_{0}(r, f), m_{0}(r, f), N_{0}(r, f)$, the readers can refer to ^{[4,5,6,7,13,14,15]}. Here, let $f, g, \alpha$ be meromorphic functions on the annulus $\mathbb{A} = \{z:\frac{1}{R_{0}} < |z| < R_{0}\},$ where $1 < R_{0}\leq+\infty.$ Then, $\alpha$ is named as a small function with respect to $f$ on $\mathbb{A}$ providing that $T_{0}(r, \alpha) = o(T_{0}(r, f))$ as $r \rightarrow \infty$ except for the set $\bigtriangleup_{r}$ such that $\int_{\bigtriangleup_{r}} r^{\lambda-1}dr < +\infty$ for $R_{0} = +\infty$, or $T_{0}(r, \alpha) = o(T_{0}(r, f))$ as $r \rightarrow R_{0}$ except for the set $\bigtriangleup^{'}_{r}$ such that $\int_{\bigtriangleup^{'}_{r}}\frac{dr}{(R_{0}-r)^{\lambda+1}} < +\infty$ for $R_{0} < +\infty$. For a nonconstant meromorphic function $f$ on the annulus $\mathbb{A}$, it is called as a transcendental meromorphic function on the annulus $\mathbb{A}$ if

or

respectively. Therefore, for a transcendental meromorphic function on the annulus $\mathbb{A}$, $S(r, f) = o(T_{0}(r, f))$ holds for all $1 < r < R_{0}$ except for the set $\bigtriangleup_{r}$ such that $\int_{\bigtriangleup_{r}} r^{\lambda-1}dr < +\infty$ or the set $\bigtriangleup^{'}_{r}$ such that $\int_{\bigtriangleup^{'}_{r}}\frac{dr}{(R_{0}-r)^{\lambda+1}} < +\infty$, respectively. Additionally we denote by $\overline{N}_{C}(r, \alpha)$ ($\overline{N}_{D}(r, \alpha)$) the reduced counting function of common zeros (different zeros) of $f-\alpha$ and $g-\alpha$ on $\mathbb{A}.$ Then, it is obvious that $f$ and $g$ share $\alpha$ IM if $\overline{N}_{D}(r, \alpha) = 0.$ Furthermore, we say $f$ and $g$ share $\alpha$ "IM'' provided that $\overline{N}_{D}(r, \alpha) = o(T_{0}(r, f))+o(T_{0}(r, g)).$

Recently, T. B. Cao, H. X. Yi and H. X. Xu ^[1,2] obtained the following five-value theorem on the annulus $\mathbb{A}$:

Theorem A ^[1,2] Let $f$ and $g$ be two transcendental or admissible meromorphic functions on the annulus $\mathbb{A} = \{z: \frac{1}{R_{0}} < |z| < R_{0}\},$ where $1 < R_{0}\leq+\infty.$ Let $a_{j}$ $(j = 1, 2, 3, 4, 5)$ be five distinct complex numbers in ${\mathbb{C}}\bigcup \{\infty\}.$ If $f$ and $g$ share $a_j$ IM for $j = 1, 2, 3, 4, 5$, then $f\equiv g.$ (In fact, this result for the case $R_{0} = +\infty$ was proved by A. A. Kondratyuk and I. Laine ^[6]).

In 2015, N. Wu and Q. Ge ^[12] further proved the five small functions theorem on the annulus $\mathbb{A}$ as follows:

Theorem B ^[12] Let $f$ and $g$ be two transcendental or admissible meromorphic functions on the annulus $\mathbb{A} = \{z: \frac{1}{R_{0}} < |z| < R_{0}\},$ where $1 < R_{0}\leq+\infty.$ Let $a_{j} (j = 1, 2, 3, 4, 5)$ be five distinct small functions with respect to $f$ and $g$ on the annulus $\mathbb{A}$. If $f$ and $g$ share $a_j$ IM for $j = 1, 2, 3, 4, 5$, then $f\equiv g$.

Naturally, it is an interesting question to investigate whether Theorem B holds if $f$ and $g$ share less than five small functions. In this paper, we mainly deal with this question, and propose the following theorems, which partly generalize the five value theorem and the five small functions theorem on annuli.

Theorem 1.1. Let $f$ and $g$ be two transcendental meromorphic functions on the annulus $\mathbb{A} = \{z:\frac{1}{R_{0}} < |z| < R_{0}\},$ where $1 < R_{0}\leq+\infty$. Let $a_{i} \equiv a_{i}(z)(i = 1, 2, 3, 4, 5)$ be five distinct small functions with respect to $f$ and $g$ on $\mathbb{A}$. If $f$ and $g$ share $a_{i}$$(i = 1, 2, 3, 4)$ "IM" and

then $f\equiv g$, where $\overline{N}_{C}(r, a_{5})$ is the reduced counting function of the common zeros of $f-a_{5}$ and $g-a_{5}$ (ignoring multiplicities) on $\mathbb{A}$.

Theorem 1.2. Let $f$ and $g$ be two transcendental meromorphic functions on the annulus $\mathbb{A} = \{z:\frac{1}{R_{0}} < |z| < R_{0}\},$ where $1 < R_{0}\leq+\infty$. Let $a_{i} \equiv a_{i}(z)(i = 1, 2, 3, 4, 5)$ be five distinct small functions with respect to $f$ and $g$ on $\mathbb{A}$. If $f$ and $g$ share $a_{i}$$(i = 1, 2, 3)$ "IM" and

then $f\equiv g$, where $\overline{N}_{D}(r, a_{j})$ are the reduced counting functions of the different zeros of $f-a_{j}$ and $g-a_{j}$ on $\mathbb{A}$.

2.   Preliminaries

In 2005, A. Y. Khrystiyanyn and A. A. Kondratyuk ^[4,5] proposed the following properties of meromorphic functions on annuli:

Lemma 2.1. ^[4] Let $f$ be a non-constant meromorphic function on the annulus $\mathbb{A} = \{z: \frac{1}{R_{0}} < |z| < R_{0}\},$ where $1 < r < R_{0}\leq+\infty,$ then the following properties always hold:

(ⅰ) $T_{0}(r, f) = T_{0}\left(r, \frac{1}{f}\right)$,

(ⅱ) $\max\{T_{0}(r, f_{1}\cdot f_{2}), T_{0}(r, f_{1}/ f_{2}), T_{0}(r, f_{1}+f_{2})\}\leq T_{0}(r, f_{1})+T_{0}(r, f_{2})+O(1)$,

(ⅲ) $T_{0}\left(r, \frac{1}{f-a}\right) = T_{0}(r, f)+O(1), \; for\; \; every\; \; fixed\; \; a\in \mathbb{C}$.

Lemma 2.2. ^[5] Let $f$ be a non-constant meromorphic function on the annulus $\mathbb{A} = \{z: \frac{1}{R_{0}} < |z| < R_{0}\},$ where $R_{0}\leq+\infty,$ and let $\lambda\geq0$. Then

(ⅰ) if $R_{0} = +\infty,$ then $m_0\left(r, \frac{f'}{f}\right) = O(\log (rT_{0}(r, f)))$ for $R\in(1; +\infty)$ except for the set $\bigtriangleup_{r}$ such that $\int_{\bigtriangleup_{r}} r^{\lambda-1}dr < +\infty;$

(ⅱ) if $R_{0} < +\infty$, then $m_0\left(r, \frac{f'}{f}\right) = O\left(\log\left(\frac{ T_{0}(r, f)}{R_{0}-r}\right)\right)$ for $r\in(1; R_{0})$ except for the set $\bigtriangleup^{'}_{r}$ such that $\int_{\bigtriangleup^{'}_{r}}\frac{dr}{(R_{0}-r)^{\lambda+1}} < +\infty.$

In addition, A. Y. Khrystiyanyn and A. A. Kondratyuk ^[5] proved the second fundamental theorem on annuli. Furthermore, T. B. Cao, H. X. Yi and H. Y. Xu ^[2] provided another form of the second fundamental theorem on the the annulus $\mathbb{A}$:

Lemma 2.3. ^[2] Let $f$ be a non-constant meromorphic function on the annulus $\mathbb{A} = \{z: \frac{1}{R_{0}} < |z| < R_{0}\}$ in which $1 < R_{0}\leq+\infty.$ Let $a_{1},$ $a_{2},$ $\ldots,$ $a_{q}$ be $q$ distinct complex numbers in the extended complex plane $\overline{\mathbb{C}}$. Then

Motivated and inspired by the ideas of ^[3,8,17], we propose the following lemmas.

Lemma 2.4. Let $f$ be a transcendental meromorphic function on $\mathbb{A} = \{z: \frac{1}{R_{0}} < |z| < R_{0}\}$ in which $1 < R_{0}\leq+\infty,$ and let $a_{1} \equiv a_{1}(z)$ and $a_{2} \equiv a_{2}(z)$ be two distinct small functions with respect to $f$ on $\mathbb{A}$.

Set

then, we have

where k = 0, 1.

Proof. It follows from the determinant nature that

This implies

by applying Lemma 2.2.

Next we can deduce

by some simple computing. It follows that

Lemma 2.4 is proved.

Lemma 2.5. Let $f$ and $g$ be two transcendental meromorphic functions on $\mathbb{A}$, and let $a_{1} = 0$, $a_{2} = 1$, $a_{3} = \infty$, $a_{4} = a(z)$ be four distinct small functions respect to $f$ and $g$ on $\mathbb{A}$, in which $a(z) \not\equiv 0, 1, \infty.$ Set

Then we get

where $\overline{N}_{D}(r, a_{i})$ is the reduced counting function of the different zeros of $f-a_{i}$ and $g-a_{i}$ on $\mathbb{A}.$

Proof. Here, we consider the counting function $N_{0}(r, H)$. It is obvious that the poles of $H$ only come from the zeros, 1-points, poles of $f$ or $g$, and the zeros of $f-a$ or $g-a$ on $\mathbb{A}$. Firstly, let $z_{4}$ be a common zero of $f-a$ and $g-a$ on $\mathbb{A}$ with multiplicity $p$ and $q$ respectively, satisfying that $a(z_{4})\neq 0, 1, \infty$. Applying the determinate nature, we have

It follows that $H(z_{4})\neq \infty$ since $z_{4}$ is a zero of $(f-g)$, and a simple pole or an analytic point of

Secondly, let $z_{3}$ (resp. $z_{1}$, $z_{2}$) be a common pole (resp. zero, 1-points) of $f$ and $g$ on $\mathbb{A}$ with multiplicity $p$ and $q$ respectively, satisfying that $a(z_{4})\neq 0, 1, \infty$ (resp. $a(z_{1})\neq 0, 1, \infty$, $a(z_{2})\neq 0, 1, \infty$). Without loss of generality, assume that $p\geq q$. By simple computation, we can write $H$ as

Noting that $z_{3}$ is a pole of $a(a-1)f'g'(f-g)^{2}$ with multiplicity $3p+q+2$ at most, a pole of $a'(f-g)[f(f-1)(g-a)g'-g(g-1)(f-a)f']$ with multiplicity $3p+2q+1$ at most, and a pole of $f(f-1)(f-a)g(g-1)(g-a)$ with multiplicity $3p+3q$, we obtain $H(z_{3})\neq \infty$. In the same manner, we can get $H(z_{1})\neq \infty, H(z_{2})\neq \infty$. Therefore, the poles of $H$ only come from the different zeros of $f, g, f-1, g-1, f-a, g-a$ and the different poles of $f, g$ on $\mathbb{A}$. In order to analyze these different zeros and different poles, we distinguish the following distinct cases.

Case 1. Let $\zeta_{1}$ be a zero of $f,$ which is neither a zero of $g,$ $a,$ and $a-1$ nor a pole of $a.$ Then, from (2.1) and (2.2) we find that $\zeta_{1}$ is a pole of $H$ with multiplicity at most $1$ if $g(\zeta_{1})\neq 1, \infty, a(\zeta_{1});$ and otherwise $\zeta_{1}$ is a pole of $H$ with multiplicity at most $2.$

Case 2. Let $\zeta_{2}$ be a zero of $f-1,$ which is neither a zero of $g-1,$ $a,$ and $a-1$ nor a pole of $a.$ By (2.1) and (2.2) we know that $\zeta_{2}$ is a pole of $H$ with multiplicity at most $1$ if $g(\zeta_{2})\neq 0, \infty, a(\zeta_{2});$ and otherwise $\zeta_{2}$ is a pole of $H$ with multiplicity at most $2.$

Case 3. Let $\zeta_{3}$ be a pole of $f,$ which is neither a pole of $g$ and $a$ nor a zero of $a$ and $a-1.$ Then, it is clear that $\zeta_{3}$ is a pole of $H$ with multiplicity at most $1$ if $g(\zeta_{3})\neq 0, 1, a(\zeta_{3});$ and otherwise $\zeta_{3}$ is a pole of $H$ with multiplicity at most $2.$

Case 4. Let $\zeta_{4}$ be a zero of $f-a,$ which is neither a zero of $g-a,$ $a,$ and $a-1$ nor a pole of $a.$ It is obvious that that $\zeta_{4}$ is a pole of $H$ with multiplicity at most $1$ if $g(\zeta_{4})\neq 0, 1, \infty;$ and otherwise $\zeta_{4}$ is a pole of $H$ with multiplicity at most $2.$

In view of the discussion above, we deduce that

Moreover, it is a direct consequence of Lemma 2.4 that $m_{0}(r, H) = S(r, f)+S(r, g)$, which implies that

Lemma 2.5 is proved.

3.   The proof of theorem 1.1

By Lemma 2.1 and Lemma 2.3, we derive that $T_{0}(r, f)\leq 3T_{0}(r, g)+S(r, f)$ and $T_{0}(r, g)\leq 3T_{0}(r, f)+S(r, g)$ noting that $f$ and $g$ share $a_{i}(i = 1, 2, 3, 4)$ "IM". Then, it is obvious that $S(r, f) = S(r, g)$.

By applying the quasi-M$\ddot{o}$bius transformation

we can assume that $a_{1} (z) = 0$, $a_{2} (z) = 1$, $a_{3} (z) = \infty$, $a_{4} (z) = a(z)$, $a_{5} (z) = b(z)$, where $a, b$ are two distinct small functions of $f$ and $g$ on $\mathbb{A}$ satisfying $a, b \not\equiv 0, 1, \infty.$ As in Lemma 2.5, we set

Since $f$ and $g$ share $0, 1, \infty, a$ "IM", we can deduce $T_{0}(r, H) = S(r, f)$ by the virtue of Lemma 2.5.

It is obvious that a common zero of $f-b$ and $g-b$ must be a zero of $H$ when it is not a zero of $b, b-1, b-a$. We assume that $H \not\equiv 0$, then, we get

This contradict $\overline{N}_{C}(r, b)\neq S(r, f)$. It follows that $H \equiv 0$.

In the following we assume that $f\not\equiv g$. From $H \equiv 0$ we have

It follows that

If $a$ is a constant, then from (3.1) we get

which further yields $f \equiv g$. This is a contradiction, we consequently have $a' \not\equiv 0$. Note that the equation (3.1) can be written as

This implies

Since $\overline{N}_{C}(r, b)\neq S(r, f)$, there exist a point $z_{0}$ satisfying that $z_{0}$ is a common zero of $f-b$ and $g-b$, but not a zero or pole of $a, a', b, b-1, b-a$. It follows that $z_{0}$ is a simple pole of the left side of (3.2), but not a pole of the right side of (3.2), which is impossible. Hence Theorem 1.1 is proved.

4.   The proof of theorem 1.2

To the contrary, we suppose that $f\not\equiv g.$ Similarly to the proof of Theorem 1.1, we can get

by utilizing Lemma 2.5. If $H \not\equiv 0$, then we have

which contradict $\overline{N}_{C}(r, b)- \overline{N}_{D}(r, a)\neq S(r, f)$. We consequently obtain $H \equiv 0$, and then the equations (3.1) and (3.2) still hold.

Note that $\overline{N}_{C}(r, b)- \overline{N}_{D}(r, a)\neq S(r, f)$ implies $\overline{N}_{C}(r, b)\neq S(r, f)$. So there exists a point $z_{0}$ satisfying that $z_{0}$ is a common zero of $f-b$ and $g-b$, but not a zero or pole of $a, a', b, b-1, b-a$. Clearly, $z_{0}$ is a simple pole of the left side of (3.2), but not a pole of the right side of (3.2). This is impossible. Therefore we have proved Theorem 1.2.

Acknowledgments

This work was supported by the NSF of China (11271227, 11561033, 61373174) and the Fundamental Research Funds for the Central Universities (JB180708). The authors would like to thank the editors and referees for making valuable suggestions to improve this paper.

Conflict of interest

The authors declare no conflicts of interest.